text: In geometric ferroelectrics, the driving force for the structural phase transition leading to the polar ferroelectric state is a rotational distortion of the polyhedra rather than an electron-sharing covalent bond formation. Such rotational distortions occur in many transition-metal oxides; in the perovskites for example they are common when the A-site cation is small, so that the oxygen octahedra collapse around it. In perovskites, the three-dimensional connectivity of the polyhedra means that no net polarization results; if one octahedron rotates to the right, its connected neighbor rotates to the left and so on. In layered materials, however, such rotations can lead to a net polarization.
  The prototypical geometric ferroelectrics are the layered barium transition metal fluorides, BaMF, M=Mn, Fe, Co, Ni, Zn, which have a ferroelectric transition at around 1000K and a magnetic transition to an antiferromagnetic state at around 50K. Since the distortion is not driven by a hybridisation between the d-site cation and the anions, it is compatible with the existence of magnetism on the B site, thus allowing for multiferroic behavior.
  A second example is provided by the family of hexagonal rare earth manganites (h-RMnO with R=Ho-Lu, Y), which have a structural phase transition at around 1300 K consisting primarily of a tilting of the MnO bipyramids. While the tilting itself has zero polarization, it couples to a polar corrugation of the R-ion layers which yields a polarisation of ~6µC/cm. Since the ferroelectricity is not the primary order parameter it is described as improper. The multiferroic phase is reached at ~100K when a triangular antiferromagnetic order due to spin frustration arises.